Cyclomatic Number Research Articles

Degree-based function index of graphs with given bipartition and small cyclomatic number

Degree-based function index of graphs with given bipartition and small cyclomatic number

Edge Laplacians and Edge Poisson Transforms for Graphs

Abstract For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantum-classical correspondence. In contrast to previously established quantum-classical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2-colorability. The quantum-classical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.

Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching

The eigenvalue multiplicity of line graphs

The eigenvalue multiplicity of line graphs

Bounds of nullity for complex unit gain graphs

Bounds of nullity for complex unit gain graphs

Omega Invariant of Complement Graphs and Nordhaus-Gaddum Type Results.

The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

New Relations Between Zagreb Indices and Omega Invariant.

In this work, we studied the problem of determining the values of the Zagreb indices of all the realizations of a given degree sequence. We first obtained some new relations between the first and second Zagreb indices and the forgotten index sometimes called the third Zagreb index. These relations also include the triangular numbers, order, size, and the biggest vertex degree of a given graph. As the first Zagreb index and the forgotten index of all the realizations of a given degree sequence are fixed, we concentrated on the values of the second Zagreb index and studied several properties including the effect of vertex addition. In our calculations, we make use of a new graph invariant, called omega invariant, to reach numerical and topological values claimed in the theorems. This invariant is closely related to Euler characteristic and the cyclomatic number of graphs. Therefore this invariant is used in the calculation of some parameters of the molecular structure under review in terms of vertex degrees, eccentricity, and distance.

On the graphs of a fixed cyclomatic number and order with minimum general sum-connectivity and Platt indices

On the graphs of a fixed cyclomatic number and order with minimum general sum-connectivity and Platt indices

The nullity of the net Laplacian matrix of a signed graph

The net Laplacian matrix of a signed graph \(\Gamma = (G, \sigma)\), where \(G = (V(G),E(G))\) is an unsigned graph (referred to as the underlying graph) and \(\sigma: E(G) \rightarrow \{-1, +1\}\) is the sign function, is defined as \(L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)\). Here, \(D^{\pm}(\Gamma)\) and \(A(\Gamma)\) represent the diagonal matrix of net-degrees and the adjacency matrix of \(\Gamma\), respectively. The nullity of \(L^{\pm}(\Gamma)\), denoted as \(\eta (L^{\pm} (\Gamma))\), refers to the multiplicity of \(0\) as an eigenvalue of \(L^{\pm}(\Gamma)\). In this paper, we concentrate on the nullity of the net Laplacian matrix of a connected signed graph \(\Gamma\), and establish that \(1 \leq \eta (L^{\pm} (\Gamma)) \leq \min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}\), where \(\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1\) denotes the cyclomatic number of \(\Gamma\). We completely determine the connected signed graphs with nullity \(|V(\Gamma)| - 1\). Additionally, we characterize the signed cactus graphs with nullity \(1\) or \(\beta(\Gamma) + 1\).

Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices

Eigenvalue multiplicity of graphs with given cyclomatic number and given number of quasi-pendant vertices

A characterization of trees with eigenvalue multiplicity one less than their number of pendant vertices

A characterization of trees with eigenvalue multiplicity one less than their number of pendant vertices

A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

Abstract Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [18] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.

Lower bounds on the general first Zagreb index of graphs with low cyclomatic number

Lower bounds on the general first Zagreb index of graphs with low cyclomatic number

The Zero Forcing Number of Graphs with the Matching Number and the Cyclomatic Number

The Zero Forcing Number of Graphs with the Matching Number and the Cyclomatic Number

Nullities of cycle-spliced bipartite graphs

For a simple graph $G$, let $\eta(G)$ and $c(G)$ be the nullity and the cyclomatic number of $G$, respectively. A cycle-spliced bipartite graph is a connected graph in which every block is an even cycle. It was shown by Wong et al. (2022) that for every cycle-spliced bipartite graph $G$, $0\leq\eta(G)\leq c(G)+1$. Moreover, the extremal graphs with $\eta(G) = c(G)+1$ and $\eta(G) =0$, respectively, have been characterized. In this paper, we prove that there is no cycle-spliced bipartite graphs $G$ of any order with nullity $\eta(G)=c(G)$. Furthermore, we also provide a structural characterization on cycle-spliced bipartite graphs $G$ with nullity $\eta(G)=c(G)-1$.

Evaluating Roads Network Connectivity for Two Municipalities in Baghdad-Iraq

The road network serves as a hub for opportunities in production and consumption, resource extraction, and social cohabitation. In turn, this promotes a higher standard of living and the expansion of cities. This research explores the road network's spatial connectedness and its effects on travel and urban form in the Al-Kadhimiya and Al-Adhamiya municipalities. Satellite images and paper maps have been employed to extract information on the existing road network, including their kinds, conditions, density, and lengths. The spatial structure of the road network was then generated using the ArcGIS software environment. The road pattern connectivity was evaluated using graph theory indices. The study demands the abstraction and examination of the topological structure by choosing a few factors associated with the connection of the roads. These involved the cyclomatic number, Eta coefficient, Aggregate Transform Score (ATS), Beta, gamma, and Alpha indices. According to the findings, the Al-Adhamiya roads network is more developed, better linked, and has a higher overall connectivity value than the Al-Kadhimiya network. The two study areas, however, have minimal circuitry and high complexity. Due to the modifications and expansion of land use that the municipalities have seen, the research suggests that the transportation network should be developed to reach greater interconnectedness, particularly in locations outside the city center.

Measuring the Urban Road network pattern using connectivity analysis: A micro-level study in Egra Municipality, West Bengal

Egra Municipality is most important for providing administrative facilities to the Purba Medinipur District. Many other facilities, such as educational facilities (school, college), super specialty hospital facilities, transport and communication facilities, etc., are attracting many people from different places. So they have chosen Egra Municipality as their destination. The central areas of this municipality are already overcrowded, and population and building densities are relatively high. Now a days, gradually, the existing administrative boundary is being extended towards the periphery. So, the transport communication system and basic amenity facilities have to be improved. The aim of this paper is to analyse the road connectivity and network accessibility of the Egra municipality. The alpha index, beta index, gamma index, and cyclomatic number have also been used for understanding network accessibility in this area. The data sets were collected through a field survey that visited 14 wards of the Egra municipality area, and secondary data was collected from the Egra municipality office and an open street map. We used the connectivity analysis method to examine the accessibility of this area. According to calculations, mainly the 7, 11, and 13 wards are more efficient than other wards. Therefore, it would suggest that there is a need to improve the connectivity and accessibility of those periphery areas for the future growth and development of the Egra municipality.

Sharp bounds for the general Randić index of graphs with fixed number of vertices and cyclomatic number

<abstract><p>The cyclomatic number, denoted by $ \gamma $, of a graph $ G $ is the minimum number of edges of $ G $ whose removal makes $ G $ acyclic. Let $ \mathscr{G}_{n}^{\gamma} $ be the class of all connected graphs with order $ n $ and cyclomatic number $ \gamma $. In this paper, we characterized the graphs in $ \mathscr{G}_{n}^{\gamma} $ with minimum general Randić index for $ \gamma\geq 3 $ and $ 1\leq\alpha\leq \frac{39}{25} $. These extend the main result proved by A. Ali, K. C. Das and S. Akhter in 2022. The elements of $ \mathscr{G}_{n}^{\gamma} $ with maximum general Randić index were also completely determined for $ \gamma\geq 3 $ and $ \alpha\geq 1 $.</p></abstract>

Inverse degree index of graphs with a given cyclomatic number

We investigate how the inverse degree index of graphs depends on their cyclomatic number. In particular, we provide sharp lower bounds on the inverse degree index over all graphs on a given number of vertices with a given cyclomatic number. We also deduce some structural properties of extremal graphs. Some open questions regarding the upper bound over the same class of graphs are discussed and some possible further developments are indicated.

Асимптотика числа помеченных планарных тетрациклических и пентациклических графов

A connected graph with a cyclomatic number k is said to be a k-cyclic graph. We obtain the formula for the number of labelled non-planar pentacyclic graphs with a given number of vertices, and find the asymptotics of the number of labelled connected planar tetracyclic and pentacyclic graphs with n vertices as n → ∞. We prove that under a uniform probability distribution on the set of graphs under consideration, the probability that the labelled tetracyclic graph is planar is asymptotically equal to 1089/1105, and the probability that the labeled pentacyclic graph is planar is asymptotically equal to 1591/1675.